1. Quality Control Chapter 5- Control Charts for Variables
PowerPoint presentation to accompany
Besterfield
Quality Control, 8e
PowerPoints created by Rosida Coowar 1

2. Outline The Control Chart Techniques State of Introduction Control
Specifications
Process Capability
Six Sigma
Different Control Charts 2

3. Learning Objectives When you have completed this chapter you should:
Know the three categories of variation and their sources.
Understand the concept of the control chart method.
Know the purpose of variable control charts.
Know how to select the quality characteristics, the rational subgroup and the method of taking samples 3

4. Learning Objectives When you have completed this chapter you should:
Be able to calculate the central value, trial control limits and the revised control limits for Xbar and
R chart.
Be able to explain what is meant by a process in control and the various out-of-control patterns.
Know the difference between individual measurements and averages; control limits and specifications. 4

5. Learning Objectives When you have completed this chapter you should:
Know the different situations between the process spread and specifications and what can be done to correct the undesirable situation.
Be able to calculate process capability.
Know the statistical meaning of 6σ 5

6. variation
The variation concept is a law of nature in that no two natural items in any category are the same.

7. Variation The variation may be quite large and easily noticeable
The variation may be very small. It may appear that items are identical; however, precision instruments will show difference
The ability to measure variation is necessary before it can be controlled

8. Variation Within-piece variation: Surface
There are three categories of variation in piece part production:
Within-piece variation: Surface
Piece-to-piece variation: Among pieces produced at the same time
Time-to-time variation: Difference in product produced at different times of the day

9. Variation Sources of Variation in production processes: INPUTS PROCESS
Measurement
Instruments
Operators
Methods
Materials
INPUTS
PROCESS
OUTPUTS
Tools
Human
Inspection
Performance
Machines
Environment

10. Variation Equipment: Toolwear Machine vibration
Sources of variation are:
Equipment:
Toolwear
Machine vibration
Electrical fluctuations etc.
Material
Tensile strength
Ductility
Thickness
Porosity etc.

11. Variation Environment Temperature Light Radiation Humidity etc.
Sources of variation are:
Environment
Temperature
Light
Radiation
Humidity etc.
Operator
Personal problem
Physical problem etc.

12. Variation
There is also a reported variation which is due to the inspection activity.
Variation due to inspection should account for one tenth of the four other sources of variation.

13. Variation
Variation may be due to chance causes (random causes) or assignable causes.
When only chance causes are present, then the process is said to be in a state of statistical control. The process is stable and predictable.

14. Control Charts Variable data x-bar and R-charts x-bar and s-charts
Charts for individuals (x-charts)
Attribute data
For “defectives” (p-chart, np-chart)
For “defects” (c-chart, u-chart)

15. Control Charts Continuous Numerical Data
Categorical or Discrete Numerical Data
Control
Charts
Variables
Attributes
Charts
Charts
This slide simply introduces the various types of control charts. R X P C
Chart
Chart
Chart
Chart

16. Control Charts for Variables
The control chart for variables is a means of visualizing the variations that occur in the central tendency and the mean of a set of observations. It shows whether or not a process is in a stable state.

17. Figure 5-1 Example of a control chart
Control Charts
Figure 5-1 Example of a control chart

18. Figure 5-1 Example of a method of reporting inspection results
Control Charts
Figure 5-1 Example of a method of reporting inspection results

19. Variable Control Charts
The objectives of the variable control charts are:
For quality improvement
To determine the process capability
For decisions regarding product specifications
For current decisions on the production process
For current decisions on recently produced items

20. Control Chart Techniques
Procedure for establishing a pair of control charts for the average Xbar and the range R:
Select the quality characteristic
Choose the rational subgroup
Collect the data
Determine the trial center line and control limits
Establish the revised central line and control limits
Achieve the objective

21. Quality Characteristic
The Quality characteristic must be measurable.
It can expressed in terms of the seven basic units:
Length
Mass
Time
Electrical current
Temperature
Substance
Luminosity
as appropriate.

22. Rational Subgroup
A rational subgroup is one in which the variation within a group is due only to chance causes.
Within-subgroup variation is used to determine the control limits.
Variation between subgroups is used to evaluate long-term stability.

23. Rational Subgroup
There are two schemes for selecting the subgroup samples:
Select subgroup samples from product or service produced at one instant of time or as close to that instant as possible
Select from product or service produced over a period of time that is representative of all the products or services

24. Rational Subgroup
The first scheme will have a minimum variation within a subgroup.
The second scheme will have a minimum variation among subgroups.
The first scheme is the most commonly used since it provides a particular time reference for determining assignable causes.
The second scheme provides better overall results and will provide a more accurate picture of the quality.

25. Subgroup Size
As the subgroup size increases, the control limits become closer to the central value, which make the control chart more sensitive to small variations in the process average
As the subgroup size increases, the inspection cost per subgroup increases
When destructive testing is used and the item is expensive, a small subgroup size is required

26. Subgroup Size
From a statistical basis a distribution of subgroup averages are nearly normal for groups of 4 or more even when samples are taken from a non-normal distribution
When a subgroup size of 10 or more is used, the s chart should be used instead of the R chart.
See Table 5-1 for sample sizes

27. Data Collection
Data collection can be accomplished using the type of figure shown in Figure 5-2.
It can also be collected using the method in Table 5-2.
It is necessary to collect a minimum of 25 subgroups of data.
A run chart can be used to analyze the data in the development stage of a product or prior to a state of statistical control

28. Figure 5-4 Run Chart for data of Table 5-2

29. Trial Central Lines
Central Lines are obtained using:

30. Trial Control Limits
Trial control limits are established at ±3 standard deviations from the central value

31. Trial Control Limits
In practice calculations are simplified by using the following equations where A2,D3 and D4 are factors that vary with the subgroupsize and are found in Table B of the Appendix.

32. Trial Control Limits
Figure 5-5 Xbar and R chart for preliminary data with trial control limits

33. Revised Central Lines

34. Standard Values

35. Figure 5-6 Trial control limits and revised control limits for Xbar and R charts

36. Figure 5-7 Continuing use of control charts, showing improved quality
Achieve the Objective
Figure 5-7 Continuing use of control charts, showing improved quality

37. Revised Central Lines

38. Sample Standard Deviation Control Chart
For subgroup sizes >=10, an s chart is more accurate than an R Chart. Trial control limits are given by:

39. Revised Limits for s chart

40. State of Control Process in Control
When special causes have been eliminated from the process to the extent that the points plotted on the control chart remain within the control limits, the process is in a state of control
When a process is in control, there occurs a natural pattern of variation

41. Figure 5-9 Natural pattern of variation of a control chart
State of Control
Figure 5-9 Natural pattern of variation of a control chart

42. State of Control Types of errors:
Type I, occurs when looking for a special cause of variation when in reality a common cause is present
Type II, occurs when assuming that a common cause of variation is present when in reality there is a special cause

43. State of Control When the process is in control:
Individual units of the product or service will be more uniform
Since the product is more uniform, fewer samples are needed to judge the quality
The process capability or spread of the process is easily attained from 6ơ
Trouble can be anticipated before it occurs

44. State of Control When the process is in control:
The % of product that falls within any pair of values is more predictable
It allows the consumer to use the producer’s data
It is an indication that the operator is performing satisfactorily

45. Common Causes
Special Causes 45

46. State of Control
Figure 5-11 Frequency Distribution of subgroup averages with control limits

47. State of Control
When a point (subgroup value) falls outside its control limits, the process is out of control.
Out of control means a change in the process due to a special cause. A process can also be considered out of control even when the points fall inside the 3ơ limits

48. State of Control
It is not natural for seven or more consecutive points to be above or below the central line.
Also when 10 out of 11 points or 12 out of 14 points are located on one side of the central line, it is unnatural.
Six points in a row are steadily increasing or decreasing indicate an out of control situation

49. Out-of-Control Condition
Change or jump in level.
Trend or steady change in level
Recurring cycles
Two populations (also called mixture)
Mistakes

50. Patterns in Control Charts
This chart enables you to discuss some of the information which can be obtained from the Process Control Charts.
Figure 5-12 Some unnatural runs-process out of control

51. Patterns in Control Charts
This chart enables you to discuss some of the information which can be obtained from the Process Control Charts.
Figure 5-13 Simplified rule for out-of-control pattern

52. Out-of-Control Patterns
Change or jump in level
Trend or steady change in level
This chart enables you to discuss some of the information which can be obtained from the Process Control Charts.
Recurring cycles
Two populations

53. Specifications
Figure 5-18 Comparison of individual values compared to averages

54. Specifications
Calculations of the average for both the individual values and for the subgroup averages are the same. However the sample standard deviation is different.

55. Central Limit Theorem
If the population from which samples are taken is not normal, the distribution of sample averages will tend toward normality provided that the sample size, n, is at least 4. This tendency gets better and better as the sample size gets larger. The standardized normal can be used for the distribution averages with the modification.

56. Figure 5-19 Illustration of central limit theorem

57. Figure 5-20 Dice illustration of central limit theorem

58. Control Limits & Specifications
Figure 5-21 Relationship of limits, specifications, and distributions

59. Control Limits & Specifications
The control limits are established as a function of the average
Specifications are the permissible variation in the size of the part and are, therefore, for individual values
The specifications or tolerance limits are established by design engineers to meet a particular function

60. Process Capability & Tolerance
The process spread will be referred to as the process capability and is equal to 6σ
The difference between specifications is called the tolerance
When the tolerance is established by the design engineer without regard to the spread of the process, undesirable situations can result

61. Process Capability & Tolerance
Three situations are possible:
Case I: When the process capability is less than the tolerance 6σ<USL-LSL
Case II: When the process capability is equal to the tolerance 6σ=USL-LSL
Case III: When the process capability is greater than the tolerance 6σ >USL-LSL

62. Process Capability & Tolerance
Case I: When the process capability is less than the tolerance 6σ<USL-LSL
Figure 5-24 Case I 6σ<USL-LSL

63. Process Capability & Tolerance
Case II: When the process capability is less than the tolerance 6σ=USL-LSL
Figure 5-24 Case I 6σ=USL-LSL

64. Process Capability & Tolerance
Case III: When the process capability is less than the tolerance 6σ>USL-LSL
Figure 5-24 Case I 6σ>USL-LSL

65. Process Capability
The range over which the natural variation of a process occurs as determined by the system of common causes
Measured by the proportion of output that can be produced within design specifications

66. Process Capability
This following method of calculating the process capability assumes that the process is stable or in statistical control:
Take 25 (g) subgroups of size 4 for a total of 100 measurements
Calculate the range, R, for each subgroup
Calculate the average range, RBar= ΣR/g
Calculate the estimate of the population standard deviation
Process capability will equal 6σ0

67. Process Capability
The process capability can also be obtained by using the standard deviation:
Take 25 (g) subgroups of size 4 for a total of 100 measurements
Calculate the sample standard deviation, s, for each subgroup
Calculate the average sample standard deviation, sbar = Σs/g
Calculate the estimate of the population standard deviation
Process capability will equal 6σo

68. Capability Index
Process capability and tolerance are combined to form the capability index.

69. Capability Index
The capability index does not measure process performance in terms of the nominal or target value. This measure is accomplished by Cpk.

70. Capability Index USL - LSL 6 ơo Cp = Cpk = min{ (USL- ¯X), (¯X-LSL)}
The Capability Index does not measure
process performance in terms of the
nominal or target
Cpk = min{
(USL- ¯X), (¯X-LSL)}

71. Capability Index
The Cp value does not change as the process center changes
Cp=Cpk when the process is centered
Cpk is always equal to or less than Cp
A Cpk = 1 indicates that the process is producing product that conforms to specifications
A Cpk < 1 indicates that the process is producing product that does not conform to specifications

72. Capability Index A Cp < 1 indicates that the process is not capable
A Cp=0 indicates the average is equal to one of the specification limits
A negative Cpk value indicates that the average is outside the specifications

73. Cpk Measures Cpk = negative number Cpk = zero Cpk = between 0 and 1

74. Six Sigma
Six Sigma is both a quality management philosophy and a methodology that focuses on reducing variation, measuring defects, and improving quality of products, processes and services.

75. Six Sigma Figure 5-27 Non-conformance rate when process is centered
Figure 5-28 Non-conformance rate when process is off center ±1.5σ

76. Different Control Charts
Charts for Better Operator Understanding:
Placing individual values on the chart: This technique plots both the individual values and the subgroup average. Not recommended since it does not provide much information.
Chart for subgroup sums: This technique plots the subgroup sum, ΣX, rather than the group average, Xbar.

77. Different Control Charts
Charts for Variable Subgroup Size:
Used when the sample size is not the same
Different control limits for each subgroup
As n increases, limits become narrower
As n decreases, limits become wider apart
Difficult to interpret and explain
To be avoided

78. Different Control Charts
Chart for Trends:
Used when the plotted points have an upward or downward trend that can be attributed to an unnatural pattern of variation or a natural pattern such as tool wear.
The central line is on a slope, therefore its equation must be determined.

79. Chart for Trends
Use Least Square Calculations

80. Figure 5-32 Chart for Trend
Chart for Trends
Figure 5-32 Chart for Trend

81. Chart for Moving Average and Moving Range
Value
Xbar R 44 46 54
48.00 10 38
46.00 16 49
47.00
44.33 11 45
46.67 4 31
40.67 15 55
43.67 24 37
41.00 42
44.67 18 43 6 47
44.00 5 51 8
Used when we cannot have multiple observations per time period
NOTE: n here is equal to 12, NOT 14
An example

82. Chart for Moving Average and Moving Range
Extreme readings have a greater effect than in conventional charts. An extreme value is used several times in the calculations, the number of times depends on the averaging period.

83. Chart for Median and Range
This is a simplified variable control chart.
Minimizes calculations
Easier to understand
Can be easily maintained by operators
Recommended to use a subgroup of 3, then all data is used.

84. Chart for Median and Range
For Table for A5, D5 and D6 see page 230

85. Chart for Individual values
Used when only one measurement is taken on quality characteristic
Too expensive
Time consuming
Destructive
Very few items

86. Chart for Individual Values
To use those equations, you have to use a moving range with n=2

87. Chart for Individual Values
Revised Limits: σo=0.8865Ro

88. Charts with Non-Acceptance Limits
Non-Acceptance limits have the same
Relationship to averages as specifications
have to individual values. Control Limits tell what the process is capable of doing, and reject limits tell when the product is conforming to specifications.

89. Charts with Non-Acceptance Limits
Figure 5-35 Relationship of non-acceptance limits, control limits
and specifications.